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Metabolic Control Analysis under Uncertainty: FrameworkDevelopment and Case Studies

Liqing Wang, _IInancx Birol, and Vassily HatzimanikatisDepartment of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois

ABSTRACT Information about the enzyme kinetics in a metabolic network will enable understanding of the function of thenetwork and quantitative prediction of the network responses to genetic and environmental perturbations. Despite recentadvances in experimental techniques, such information is limited and existing experimental data show extensive variation andthey are based on in vitro experiments. In this article, we present a computational framework based on the well-established(log)linear formalism of metabolic control analysis. The framework employs a Monte Carlo sampling procedure to simulate theuncertainty in the kinetic data and applies statistical tools for the identification of the rate-limiting steps in metabolic networks.We applied the proposed framework to a branched biosynthetic pathway and the yeast glycolysis pathway. Analysis of theresults allowed us to interpret and predict the responses of metabolic networks to genetic and environmental changes, and togain insights on how uncertainty in the kinetic mechanisms and kinetic parameters propagate into the uncertainty in predictingnetwork responses. Some of the practical applications of the proposed approach include the identification of drug targets formetabolic diseases and the guidance for design strategies in metabolic engineering for the purposeful manipulation of themetabolism of industrial organisms.

INTRODUCTION

For more than a century, substantial scientific efforts have

been invested in exploring the cellular metabolism to

understand the properties of its elementary components,

such as enzymes, and distinct subsystems, such as bio-

synthetic pathways. As a result, significant advancements

have been made in this field, which in turn have led to the

appreciation of the importance of studying individual en-

zymes within the context of metabolic networks and their

physiological environment (Bailey, 1991, 1998; Papin et al.,

2003). Metabolic flux analysis (MFA) is a framework that

addresses an important aspect of this problem through the

identification and analysis of the metabolic fluxes, i.e.,

steady-state reaction rates, in metabolic networks (Papout-

sakis, 1984; Vallino and Stephanopoulos, 1993; Varma and

Palsson, 1993a,b). The mass balance equations of metabolic

intermediates and the balance equations of energy and redox

allow the formulation of linear constraints on the chemical

reaction rates around each metabolite. Some of the metabolic

fluxes can be estimated through measurements of the

consumption and production rates of extracellular metabo-

lites, i.e., substrates and products, and through tracer ex-

periments with stable isotopes that allow the estimation of

some key intracellular reactions (Klapa et al., 2003; Sauer

et al., 1997; Schmidt et al., 1999). This experimental

information is used together with the linear constraints to

obtain a quantitative estimation of the metabolic fluxes.

Constraints-based analysis (Price et al., 2003; Varma and

Palsson, 1993a,b) is another MFA approach based also on the

linear constraints on themetabolic reaction rates, and it allows

the investigation of a broad range of properties of metabolic

networks, such as the flux distribution in the metabolic

network, that can support optimal growth rate, physiological

responses of the flux distribution after gene deletion, medium

requirements, and network robustness (Price et al., 2003).

MFA has been widely applied to interpret cellular

physiology as well as to design experiments for redirecting

metabolic fluxes for improved biological performance in

medical and biotechnological applications (Stephanopoulos

and Vallino, 1991; Varma and Palsson, 1993a,b; Yarmush

and Berthiaume, 1997). However, MFA is limited in its

ability to identify how fluxes in the metabolic networks are

reconfigured in response to environmental and genetic

changes since information about the kinetic properties of

individual enzymatic steps in the metabolic networks is not

considered within the analysis.

A variety of conceptual approaches have been developed to

introduce kinetic information into the study of metabolic

networks (Teusink et al., 2000; Vaseghi et al., 1999).

Metabolic control analysis (MCA), initially called metabolic

control theory, was one of the first frameworks developed

for the study of metabolic networks with respect to their

sensitivity to biochemical and environmental variations

(Kacser and Burns, 1973). MCA offers a rigorous theoretical

means for the quantification of the steady-state and dynamic

responses of fluxes and metabolite concentrations induced by

the changes of system parameters such as enzyme activities

(Hatzimanikatis and Bailey, 1997; Kacser and Burns, 1973).

Since its establishment, this conceptual framework has

undergone extensive developments (Fell and Sauro, 1985;

Submitted June 21, 2004, and accepted for publication September 22, 2004.

Address reprint requests to Vassily Hatzimanikatis, E-mail: vassily@

northwestern.edu.

_IInancx Birol’s present address is Dept. of Chemical and Environmental

Engineering, Illinois Institute of Technology, Chicago, IL 60616.

� 2004 by the Biophysical Society

0006-3495/04/12/3750/14 $2.00 doi: 10.1529/biophysj.104.048090

3750 Biophysical Journal Volume 87 December 2004 3750–3763

Hatzimanikatis and Bailey, 1996, 1997; Heinrich and

Rapoport, 1974; Kholodenko and Westerhoff, 1993; Reder,

1988) and attracted significant attention as a powerful tool

in basic biology, biophysics, biotechnology, and medicine

(Berthiaume et al., 2003; Bowden, 1999; Cascante et al.,

2002; Schuster, 1999; Westerhoff and Kell, 1996).

However, a persisting hurdle in MCA is the lack of

comprehensive knowledge of the kinetic properties of the

enzymes in a metabolic network. Although such information

is available for many enzymes (Schomburg et al., 2002) and

biochemical techniques allow measurements of the kinetic

properties of a number of enzymes (Schmidt et al., 1999;

Teusink et al., 2000), such information is generally obtained

from in vitro studies, and inmost of the cases, details about the

in vivo kinetic properties of the enzymes are not accessible. In

addition, even when the intracellular properties are inferred

through sophisticated experiments, these measurements are

subject to variations resulting from differences in experimen-

tal systems and conditions. Furthermore, most experiments

conducted on living organisms to measure their metabolic

properties yield results that are essentially ensemble aver-

ages. Thus, these results are inherently subject to extensive

variations due to the differences between individual cells.

The uncertainty in enzyme kinetic parameters and its

impact on the prediction of the metabolic properties have

attracted considerable attention. A number of approaches

have been proposed that employ parameter space sampling

procedures and statistical analysis tools to study character-

istics of metabolic network under parameter variations

(Almaas et al., 2004; Alves and Savageau, 2000; Petkov

and Maranas, 1997; Pritchard and Kell, 2002; Thomas and

Fell, 1994). In this study, we introduce a computational

framework that enables the statistical characterization of

the kinetic responses of metabolic networks. It integrates

information from biochemistry, genomics, cell physiology,

and MFA, while taking into account the uncertainty

associated with the kinetic information of enzymes in the

network. Based on a previously developed (log)linear MCA

formalism (Hatzimanikatis and Bailey, 1996, 1997; Hatzi-

manikatis et al., 1996), the number of metabolic parameters

required for the analysis are reduced to a minimum, and the

high degree of uncertainty, as well as the partial knowledge

about the kinetic properties of the enzymes in metabolic

pathways, are addressed using a Monte Carlo method that

relies on large-scale computation. Statistical analysis of the

simulation results allows us to identify and characterize

quantitatively the rate-limiting steps in two characteristic

metabolic networks: a branched biosynthetic pathway and the

glycolysis pathway in yeast.

METHODS

Mathematic modeling of metabolic networks

For any metabolic system, the dynamics of the metabolite concentrations can

be expressed by the equation that describes the metabolite mass balances,

dxdt

¼ Nvðx; pe; psÞ; (1)

where x is the metabolite concentration vector, N is the stoichiometric

matrix, v is the metabolic flux vector, pe is the enzyme activity parameter

vector, which includes both kinetic parameters and enzyme concentrations,

and ps is the vector of other system parameters such as temperature and pH

(for nomenclature and dimensions, see Table 1). Reversible fluxes are de-

composed into two separate and opposite irreversible fluxes. The values of

the metabolic fluxes are in general functions of metabolite concentrations,

enzyme kinetic parameters, and other system parameters, such as pH and

temperature. In particular, the forward and backward rate expressions of the

same enzymatic reaction share some of the kinetic parameters.

In manymetabolic networks, the concentration of some of the metabolites

are subject to conservation constraints. Common examples include the com-

pounds involved in energy (ATP, ADP, and AMP) and redox metabolism

(NAD and NADH). The total amount of each group of these compounds,

which are called conserved moieties, remains invariant over the characteristic

response time of themetabolic network, although individual compound levels

may vary (Heinrich et al., 1977; Reich et al., 1976). Consideration of the

conserved moieties requires the introduction of a third parameter vector, pm,which represents the total concentration of the metabolites in each moiety

group. Within each conserved moiety, the conservation constraint requires

that the concentration of some compounds depends on the concentration of

their counterparts. Accordingly, the original set of metabolite concentrations

x is divided into two categories: an independent metabolite concentration

vector, xi, and a dependent metabolite concentration vector, xd. Each element

of the latter belongs to a different conserved moiety.

Conservation relationships among metabolites in a metabolic network

lead to a rank deficiency of the stoichiometric matrix, N, by introducing

linearly dependent rows. The rows that correspond to the mass balances of

the independent metabolites can be extracted from the stoichiometric matrix,

N, and form a new stoichiometric matrix, NR (Heinrich and Schuster, 1996;

Reder, 1988). Consideration of the above constraints, leads to the reduction

of the mass balance equations of the metabolic network (Eq. 1) into the form

dxi

dt¼ NR vðxi; xdðxi; pmÞ; pe; psÞ; (2)

where the set of system parameters, p, consists of the conserved moiety

concentrations pm, the enzyme activity parameters pe, and other parameters

included in ps,

pT ¼ ½pTm...pTe...pTs �: (3)

(Log)linear kinetic formalism of MCA

Within the MCA framework, concentration control coefficients,Cxp; and flux

control coefficients, Cvp; are defined as the fractional change of metabolite

concentrations and metabolic fluxes, respectively, in response to fractional

changes of system parameters (Table 1) (Kacser and Burns, 1973).

Following the established (log)linear model formalism (Hatzimanikatis

et al., 1996; Reder, 1988), we can linearize and scale the system (Eq. 2)

around the steady state, and derive these expressions for the control

coefficients (see Supplementary Material for details):

Cxip ¼�ðNRVEi1NRVEdQiÞ�1½NRVPm

..

.NRVPe

..

.NRVPs�;

(4)

Cvp ¼ ðEi1EdQiÞCxi

p 1 ½Pm...Pe

..

.Ps�: (5)

Here, V is the diagonal matrix whose elements are the steady-state fluxes; Ei

and Ed are the matrices of the elasticities with respect to metabolites, defined

MCA under Uncertainty 3751

Biophysical Journal 87(6) 3750–3763

as the local sensitivities of metabolic fluxes to independent and dependent

metabolite concentrations, respectively; IIm, IIe, and IIs are the matrices of

the elasticities with respect to parameters, i.e., the local sensitivities of

metabolic fluxes to system parameters, pm, pe, and ps, correspondingly(Table 1); andQi is a weight matrix that represents the relative abundance of

dependent metabolites with respect to the abundance of the independent

ones. A second weight matrix, Qm, is also defined, for the relative

abundance of dependent metabolites with respect to the levels of their

corresponding total moieties, which leads to the expression for the matrices

of elasticities with respect to parameters, IIm,

Pm ¼EdQm: (6)

Numerical calculation of control coefficients

Eqs. 4 and 5 suggest that the values of the flux and concentration control

coefficients depend on information from only four levels: system stoichiom-

etry, flux distribution, conservedmoiety compositions, and elasticities. Eqs. 4

and 5 lead to the important observation that the control coefficients do not

depend explicitly on the concentration of the metabolites. The concentration

of the metabolites affects only the values of the elasticities and the local

dynamics of the system.

The stoichiometric matrix, N, can be constructed based on established

biochemical studies and genomic information (Forster et al., 2003; Kanehisa

and Goto, 2000; Krieger et al., 2004), and the reduced stoichiometric matrix,

NR, can be readily deduced from system stoichiometry through the

identification of the conserved moiety groups (Schuster and Hilgetag, 1995).

The values of the net steady-state fluxes in the matrix V can be estimated

based on MFA studies (Varma and Palsson, 1993a,b; Schmidt et al., 1999;

Teusink et al., 2000). Furthermore, each reversible flux is decomposed into

a forward and a backward flux. We can define the equilibrium coefficient, r e

(0,1N), as the ratio of the forward flux rate and the backward reaction rate,

r¼ vfvb: (7)

The equilibrium coefficient is a measure of the value of the reversible steps

relative to the net flux through the reversible enzyme,

r�1¼ vf � vbvb

¼ vnetvb

: (8)

TABLE 1 MCA nomenclature

3752 Wang et al.


A r-value close to 1 corresponds to forward and backward fluxes that are

much greater than the net flux and indicates that the enzyme operates near

thermodynamic equilibrium. The values r ¼ 0 and r / N correspond to

irreversible reactions in the backward and forward directions, respectively.

Fluxes in both directions can be calculated from the net flux value and the

equilibrium coefficient.

The matrices of elasticities with respect to metabolites, Ei and Ed, depend

on the local sensitivities of enzymatic reaction rates to metabolite concentra-

tions. For example, for an enzyme that follows irreversible Michaelis-

Menten kinetics,

vi ¼ vmax;i

xjKm;i1xj

; (9)

the elasticity with respect to metabolite, ei,j, can be calculated as

ei;j ¼ @ lnvi@ lnxj

¼ 1

11xj=Km;i

: (10)

Note that ei,j is a function of the scaled substrate concentration xj/Km,i only,

and is bounded between 0 and 1. More generally, it can be easily shown that

values of the elasticities with respect to metabolites for enzymes that follow

other types of common kinetics also lie within well-defined bounds (Segel,

1975) (see Supplementary Material for some examples). For reversible

reactions, as well as for reactions with multiple reactants and products, the

enzyme elasticities with respect to substrates and products are correlated,

since individual elasticities are functions of a common set of metabolite

concentrations. For example, the rate expression of a single-substrate single-

product reaction that follows reversible Michaelis-Menten kinetics can be

written as

vf ¼ vmax;f

a

11a1p; (11)

vb ¼ vmax;b

p

11a1p; (12)

where a and p represent the scaled substrate and product concentrations,

respectively, and the subscripts f and b denote the forward and backward

reactions, respectively. The elasticities of these rate expressions with respect

to substrate and product metabolite can be expressed as

ef;a ¼ 11p

11a1p; (13)

ef;p ¼ �p

11a1p; (14)

eb;a ¼ �a

11a1p; (15)

eb;p ¼ 11a

11a1p: (16)

These relationships suggest that the elasticities of this enzyme kinetic

mechanism are correlated since four elasticities depend on two metabolite

concentrations. Furthermore, the elasticities with respect to the reactant are

in the range of [0,1] (Eqs. 13 and 16) and those with respect to the product

are in the range of [�1,0] (Eqs. 14 and 15) (see Supplementary Material for

the elasticities of enzymes with multiple reactants and products in the

glycolysis pathway case study).

In most of the common rate expressions for enzyme kinetics (Segel,

1975), as in the following case studies, the reaction rate is proportional to

a parameter called maximum reaction rate, vmax, which is the product of the

total enzyme concentration and a catalytic rate constant, commonly called

maximum specific enzyme activity. Under these conditions, the correspond-

ing elasticity with respect to maximum enzyme activity is equal to 1, and

therefore the corresponding elements in the matrix of elasticities with respect

to parameters, IIe, are equal to 1. Some of the most common exceptions

include the enzyme channeling cases (Kholodenko and Westerhoff, 1993),

which will be the subject of future work.

The elements of the weight matrix, Qm, can be estimated based on

knowledge about the relative concentration of the metabolites in the con-

served moieties (see Results and Discussion, below, on how to obtain such

estimates) and the elements of the moiety elasticity matrix, IIm, can be

obtained from Eq. 6. Finally, the elements of the elasticity matrix, IIs,describe the effect of changes in other system parameters and can be

estimated through information about the enzyme kinetics or perturbation

experiments. In the case-studies considered here we do not consider changes

in these parameters, and therefore the elasticity matrix, IIs, is not consideredin this article.

Monte Carlo simulation and stability analysis

The uncertainty in the quantification of metabolic fluxes, enzyme kinetic

properties, and metabolite concentrations, does not allow a precise quanti-

fication of the control coefficients. To address this problem, we have de-

veloped a Monte Carlo sampling methodology that allows us to describe

control coefficients using statistical analysis and representation (Fig. 1).

The method consists of the following steps:

Step 1. We define the stoichiometry based on biochemistry and

genomics knowledge about the system.

Step 2. Steady-state fluxes are estimated based on experimental

measurements and MFA methods.

Step 3. Through stoichiometric analysis, we identify the conserved

moieties and we group metabolites into an independent group and

dependent group.

Step 4. We assume a distribution for the value of the scaled metabolite

concentrations that determine the distributions in the values of the

corresponding elasticities. When details about the rate expressions of

the enzymes are not available, we either use simple random genera-

tion of the elasticity values, or assume ad hoc kinetics. If the system

FIGURE 1 Diagrammatic description of the Monte Carlo simulation

algorithm.



involves conserved species, we also generate random values for the

relative individual concentrations.

Step 5. For the generated elasticity values and relative values of the

conserved moieties, we test the local stability of the steady state.

Although the values of control coefficients do not depend onmetabolite

concentration levels, for every randomly generated system, i.e., for a set

of elasticities, the metabolite concentration levels will determine the

local stability of the system. Thus, for every system in the population of

the control coefficients, we generate random values for the metabolite

concentration levels within physiological bounds, and we examine the

local stability of the system based on the eigenvalues of the Jacobian

matrix,

J¼NRVðEi1EdQiÞX�1

i ; (17)

where Xi is a diagonal matrix of the corresponding metabolite

concentrations. For feasible values of the control coefficients, all the

eigenvalues of this matrix should have negative real parts for each

sample.

Step 6. If the steady state is not stable, we discard the set of values and

return to Step 4 to generate another random system. If the steady state

is stable, we calculate the corresponding control coefficients, store

the data, and return to Step 4 for the next iteration.

After repeating this process multiple times, we generate a population of

control coefficients. Unstable configurations are rejected and only stable

ones are considered for the statistical analysis of the simulated systems.

Although stable states that could correspond to locally unstable steady states,

such as oscillations, have been observed in some metabolic systems, the

current method has been developed for systems whose observable fluxes

suggest that the system is operating at stable quasisteady states, i.e., the

fluxes remain time-invariant over the observable timescale. In such systems,

the naturally occurring condition appears to be stable and we do not

introduce any bias by rejecting unstable solutions.

Monte Carlo simulations: elasticities

Enzyme elasticities associated with most common kinetic mechanisms are

uniquely determined by the scaled metabolite concentrations, which require

a complete knowledge about in vivo metabolite concentration levels and

enzyme kinetic parameters. In general, such information is not available, and

when available, it is subject to uncertainty due to experimental conditions as

discussed above. Furthermore, in vivo conditions are very different from the

in vitro conditions under which the kinetic parameters have been estimated.

To simulate these uncertainties in the elasticities, we introduced the

following Monte Carlo sampling technique.

Most of the enzymatic reactions initiate with the binding of the substrate

or regulatory metabolite, S, on the active site, A, of the enzyme

A1S�!kon AS

AS�!koff A1S

: (18)

According to this model, the free active site of the enzyme, A, binds

reversibly to metabolite, S, to form the enzyme-substrate complex, AS, with

an on-rate constant, kon, and an off-rate constant, koff. The total active site

concentration is the sum of the concentration of the free active site and the

active site-metabolite complex,

½AT� ¼ ½A�1 ½AS�; (19)

where [�] represents the concentration value of the corresponding species.

The degree of the saturation of the active site can then be defined as

sA ¼ ½AS�½AT� ¼

½S�=Km

½S�=Km11; (20)

where

Km ¼ koffkon

; (21)

and the scaled metabolite concentration can therefore be expressed in terms

of sA as

½S�Km

¼ sA

1�sA

: (22)

Assuming that the population of the active site in each species is evenly

distributed between nonsaturation and full saturation, we uniformly sample

sA by assigning random numbers between 0 and 1. Eq. 22 provides a general

way of sampling scaled metabolite concentrations to generate random

independent samples of scaled metabolite concentrations, which we can then

use to calculate the elasticities with respect to metabolites, and subsequently,

the control coefficients. Note that the distributions of the randomized

elasticities with respect to metabolites will be determined by the particular

enzyme kinetic mechanisms. For example, for enzymes that follow

Michaelis-Menten kinetics, the elasticity with respect to the substrate will

be uniformly distributed between 0 and 1; whereas for enzymes that follow

Hill kinetics, the elasticity with respect to the substrate will be distributed

between 0 and the value of the Hill coefficient, nh, following a bimodal

distribution with peaks at the limiting values (Fig. 2). These observations are

in agreement with the nature of different kinetic mechanisms. Enzymes that

follow Hill kinetics switch between an active form and an inactive form,

depending on the saturation status. Therefore the elasticity falls into two

regimes corresponding to the different enzyme activity states. This type

of information about the enzyme kinetics is captured by the Monte

Carlo sampling and the uncertainty in the enzyme saturation levels, i.e.,

the uncertainty in both enzyme kinetics and metabolite concentrations,

is translated into uncertainty in the control coefficients, which can

be statistically quantified and analyzed. Furthermore, the bounds and

distribution of sA can be refined for systems where detailed knowledge of

the degree of enzyme saturation is available (Fig. 2).

FIGURE 2 Uniform sampling of enzyme saturation degrees and random-

ization of elasticities with respect to metabolites. Histograms of elasticities

with respect to metabolites for Michaelis-Menten kinetics and Hill kinetics

(Hill coefficient ¼ 2) for uniformly sampled enzyme saturation degree, s,

at three scenarios: uniform sampling between 0 and 1 (upper panels)corresponding to the full range of enzyme saturation degrees; uniform

sampling between 0 and 0.5 (middle panel) corresponding to low enzyme

saturation; and uniform sampling between 0.5 and 1 (lower panel)

corresponding to high enzyme saturation.

3754 Wang et al.


Monte Carlo simulations: conserved moieties

The calculation of the conserved moiety weight matrices, Qi and Qm, is

illustrated here using the two most prevalent and important conserved

moieties in living cells: the adenylates group (AMP, ADP, and ATP) and the

pyridine nucleotides group (NAD and NADH). Within each moiety, the

individual species will be interconverted to each other through reactions in

the metabolic network, but the total amount of each group always remains

constant (Andersen and von Meyenburg, 1977; Ball and Atkinson, 1975;

Reich, 1974; Reich and Selkov, 1981).

For the adenylates moiety, based on the conservation relationship

½ATP�1 ½ADP�1 ½AMP� ¼ ½AP�; (23)

any of the three cofactors can be selected as a dependent metabolite (e.g.,

AMP) and the other two (ATP and ADP) will be independent metabolites.

From the definition of Qi and Qm, their elements are merely the relative

ratios of concentration of the various species within the conserved moiety

(Table 1):

d ln ½AMP�d ln ½ATP� ¼� ½ATP�

½AMP�; (24)

d ln ½AMP�d ln ½ADP� ¼�½ADP�

½AMP�; (25)

d ln ½AMP�d ln ½AP� ¼ ½AP�

½AMP�: (26)

Uncertainty should be considered again in the analysis since intracellular

metabolite concentrations might be unknown and they usually fluctuate

under different cellular states and environments. In general, each of the

metabolite concentrations could be uniformly sampled from physiological

ranges if experimental measurements exist (Teusink et al., 2000).

For the adenylates pool, in particular, the values of the weight matrix

entries are directly connected with the energetic state of the living cell. All

three adenylates are important regulators of metabolic reactions and, more

generally, many catabolic and anabolic processes are regulated by the energy

status of the cell. An index named energy charge (ec) has been defined to

reflect cellular energy status by quantifying the relative number of high-

energy phosphate bonds in the adenylates moiety (Atkinson, 1968),

ec¼ ½ATP�10:5½ADP�½AP� : (27)

Although the value of energy charge can vary between 0 and 1, living cells

maintain its value within a small range,;0.9 in Escherichia coli and 0.8–0.9

in Saccharomyces cerevisiae (Andersen and vonMeyenburg, 1977; Ball and

Atkinson, 1975). For a given ec value, the steady-state values of intracellularadenylates concentrations have the relations of

½ADP� ¼ 23ec3½AP��2½ATP�; (28)

½AMP� ¼ ð1�23ecÞ½AP�1 ½ATP�: (29)

After substituting these relationships into Eqs. 24–26, we can express the

elements of Qi and Qm as

d ln ½AMP�d ln ½ATP� ¼� ½ATP�=½AP�

1�23ec1 ½ATP�=½AP�; (30)

d ln ½AMP�d ln ½ADP� ¼� 23ec�2½ATP�=½AP�

1�23ec1 ½ATP�=½AP�; (31)

d ln ½AMP�d ln ½AP� ¼ 1

1�23ec1 ½ATP�=½AP�: (32)

These equations suggest that the elements of the adenylates weight matrices

can be calculated based on the values for the ec index and the fraction of

ATP in the adenylates pool. The viable ranges of the ec index has been

experimentally accessed for different organism species and physiological

conditions of the cells (Ball and Atkinson, 1975; Chapman et al., 1971). Eqs.

28 and 29 also suggest that, for all adenylates concentrations to be positive,

the ATP fraction has to be bounded as

23ec�1,½ATP�½AP� ,ec: (33)

Therefore, we uniformly sample ec index values within the physiological

range and, for each ec index, we uniformly sample the relative ATP levels

between the bounds defined by Eq. 33. Thus, the corresponding indices of

the weight matrices are randomized based on these samples (Eqs. 30–32).

Applying similar concepts on the quantification of the pyridine

nucleotides moiety, we start from the linear conservation relationship

½NAD�1 ½NADH� ¼ ½AN�: (34)

If we choose NAD as the dependent metabolite, the corresponding elements

in Qi and Qm are again the relative ratios of moiety concentrations,

d ln ½NAD�d ln ½NADH� ¼�½NADH�

½NAD� ; (35)

d ln ½NAD�d ln ½AN� ¼ ½AN�

½NAD�: (36)

Taking physiological conditions into consideration, the pyridine

nucleotides must be largely oxidized for the glycolytic reactions to proceed.

An index called the catabolic reduction charge (crc) has been defined to

represent the redox status of the cellular condition (Andersen and von

Meyenburg, 1977),

crc¼ ½NADH�½AN� : (37)

After the introduction of this index, Eqs. 35 and 36 can be expressed as

d ln ½NAD�d ln ½NADH� ¼� crc

1� crc; (38)

d ln ½NAD�d ln ½AN� ¼ 1

1� crc: (39)

Therefore, we uniformly sample crc values within the physiological range,

;0.05 in aerobically grown E. coli and 0.001;0.0025 in S. cerevisiae(Andersen and von Meyenburg, 1977; Holzer et al., 1956) and get the

elements of the weight matrices.

Statistical analysis

After feasible configurations of simulation outputs are calculated and

collected, statistical properties of the control coefficients are analyzed. The

primary tool that we use in the studies presented here is that of the

complementary cumulative distribution functions (CCDFs). The comple-

mentary cumulative distribution function, denoted here by F(x), measures

the probability that the random variable X assumes a value greater or equal to

x; that is, F(x)¼ P(X$ x) (Papoulis and Pillai, 2002). For a discrete sample,

like the Monte Carlo simulation used in this study,

FðxÞ ¼ +all

xi$x

pðxiÞ: (40)

The CCDF will provide a measure of the probability that a control

coefficient is greater than a certain value. Further statistical analysis of the



populations of the control coefficients could also be performed (Pritchard

and Kell, 2002), but it is beyond the scope of this article.

RESULTS AND DISCUSSION

In this section, we will illustrate the application of the MCA

Monte Carlo method using two case studies: a prototypical

module of biosynthetic pathways; and the more complex

yeast glycolytic pathway.

Branched pathway

A primary module of metabolic networks is the branched

pathway which, together with the linear pathway, constitutes

the major building blocks of most metabolic networks. The

splitting ratios of branching fluxes at key nodes essentially

determine the overall distribution of metabolic fluxes in

most biological systems. We chose the branched pathway

presented in Fig. 3 as our first example to illustrate how to

apply our framework on actual pathway models and how to

interpret the simulation results.

Assuming irreversible Michaelis-Menten kinetics for all

five fluxes, we derived an analytical expression for the flux

control coefficients (Table 2). The expressions suggest that

the flux control coefficients depend on the enzyme elasticities

and on the splitting ratios alone. We used the sampling pro-

cedure described above and we generated populations of

control coefficients for different splitting ratios of the flux

through the pathway to study the distribution of the control

coefficient of flux through enzyme 2, v2, with respect to the

activity of enzyme 2, pe2 :Statistical analysis of the simulated results suggests that

the splitting ratio has a critical impact on the control

coefficients distributions. When the flux through enzyme 2

is a small portion of the overall flux, i.e., a splitting ratio

a ¼ v2/(v11v2) ¼ 0.1, in 95% of the cases the control

coefficients of v2 with respect to pe2ðCv2pe2) will be.0.5, with

a mean value 0.84 (Fig. 4). As the splitting ratio a increases

from 0.1 to 0.9, the ability to control v2 via manipulation of

pe2 is continuously attenuating, as indicated by decreasing

mean values of Cv2pe2

and the dramatic change in the CCDF

(Fig. 4).

Reversibility also affects the distributions of control

coefficients. If v2 is assumed to be a reversible reaction (Fig.

3 b) (see Table 3 for MCA elements of this model), and we

consider the control of pe2 over the net flux v2, the value ofthe equilibrium coefficient has a profound effect on the

FIGURE 3 Branched pathway models. Four types of branched bio-

synthetic pathways consisting of three metabolites: (a) simple irreversible

branched pathway; (b) branched pathway with a reversible reaction (n2); (c)

branched pathway with feedback inhibition; and (d) branched pathway withfeedback inhibition and crossover activation.

TABLE 2 MCA elements and analytical results for the branched pathway model with irreversible kinetics

3756 Wang et al.


magnitudes of control coefficients (Fig. 5 a). The closer

the flux v2 is to equilibrium, the smaller the control coefficient

Cv2pe2(Fig. 5 b) will be. This illustrates the well-known

postulate that enzymes that operate near thermodynamic

equilibrium do not have significant control over metabolic

fluxes. In addition, this result also captures the fact that

reversible reactions can have a significant control of meta-

bolic fluxes if they do not operate near equilibrium, contrary

to the common misunderstanding that reversible enzymes do

not have control over metabolic fluxes regardless of their

operating state with respect to the equilibrium conditions.

Another critical feature of enzyme-catalyzed reactions is

their capacity to adjust according to the metabolic require-

ments. Such control sometimes is accomplished at the

transcription level by enzyme synthesis through induction

and repression, which can involve complicated molecular

mechanisms. At the enzymes level, inhibition and/or

activation of the enzymes by the metabolites in a metabolic

network are extensively employed to regulate metabolic

fluxes. We studied how the distributions of the control

coefficients of the unregulated branched pathway change

when we consider two different regulatory structures: 1),

a product competitive inhibition scheme (Fig. 3 c); and 2),

a product competitive inhibition with crossover cooperative

activation scheme (Fig. 3 d).When product competitive inhibition is present, the

kinetic expressions for v2 and v3 are considered to be

v2 ¼ v2max

x1=Km;2

x1=Km;21x2=KI;211; (41)

v3 ¼ v3max

x1=Km;3

x1=Km;31x3=KI;311; (42)

where KI,2 and KI,3 are the inhibition constants. When

crossover cooperative activation is included with competi-

tive inhibition, the kinetics of v2 and v3 are expressed as

v2 ¼ v2max

x1=Km;2

x1=Km;21x2=KI;2113

x3=KA;2

x3=KA;211; (43)

v3 ¼ v3max

x1=Km;3

x1=Km;31x3=KI;3113

x2=KA;3

x2=KA;311; (44)

whereKA,2 andKA,3 are the activation constants. As indicated

by the results of Fig. 6 a, the regulation imposed on the

TABLE 3 MCA elements for the branched pathway model with reversible kinetics

FIGURE 4 Effects of splitting ratio on the flux control coefficients in

branched pathway. CCDFs of the flux control coefficients Cv2pe2

of the

branched pathway for different splitting ratios, a ¼ v2/v1. The CCDF

measures the probability that the random variable Cv2pe2

is $x.



enzymes enhances the rigidity of the system and consequently

attenuates the ability of controlling the flux through enzyme 2

by manipulating enzyme activity as expected. This observa-

tion is in agreement with the metabolic rigidity concept,

which states that flux alterations at branching points will be

largely constrained by control architectures of the network

(Stephanopoulos and Vallino, 1991).

Different enzyme regulation mechanisms also influence

the distributions of control coefficients. To illustrate this, we

compared two common types of activations in the inhibition-

activation branched pathway model of Fig. 3 d, i.e., co-operative activation and allosteric activation. The kinetics of

n2 and n3 with allosteric activation are expressed as

v2 ¼ v2max

x1=Km;2

11x2=KI;2

11x3=KA;2

1x1=Km;2

; (45)

v3 ¼ v3max

x1=Km;3

11x3=KI;3

11x2=KA;3

1x1=Km;3

; (46)

which are very different from the regulation mechanisms

described by Eqs. 43 and 44. Despite the differences in the

enzyme regulation mechanisms, the differences in the

distribution of the control coefficients are not dramatic,

although the difference of the means of the two populations

of the control coefficients is statistically significant (Fig. 6 b).These results also illustrate that in the absence of knowledge

about the kinetic mechanisms we can assume alternative

mechanisms, perform the Monte Carlo studies, and evaluate

the significance of the alternative mechanisms on the

calculation of the control coefficients.

In a more general case when the activation mechanism

is unknown for a given enzyme, we can randomize in-

dependently the elasticity of inhibitor uniformly within

FIGURE 5 Effects of equilibrium coefficient r on control coefficients. (a)CCDFs of the control coefficient Cv2

pe2at four different values of the equi-

librium coefficient r of reaction v2 in the branched pathway with v2/v1¼ 0.1.

The values are r ¼ N (solid line); r ¼ 1.1 (dashed line); r ¼ 2 (dottedline); and r ¼ 10 (dash-dotted line). (b) Box plot of Cv2

pe2distributions, with

middle lines representing the median of the distributions; the lower and the

upper bounds of the boxes corresponding to the first and the third quartiles;

and the dashed lines extending from each end of the box to show the range of

the data.

FIGURE 6 Impacts of enzyme regulation on the control coefficients

distributions. (a) CCDFs of control coefficient Cv2pe2

for cases of regulatory

structures in the branched pathway with v2/v1 ¼ 0.1. No enzyme regulation

(solid line); product competitive inhibition (dashed line); and product

competitive inhibition and crossover cooperative activation (dotted line). (b)CCDFs of the control coefficient Cv2

pe2for the same regulatory structure

(product competitive inhibition and crossover cooperative activation) and

for different kinetic mechanisms: (1) cooperative activation (solid line); (2)

allosteric activation (dashed line); and (3) generic activation (dotted line).

3758 Wang et al.


[�1,0] and that of activator uniformly within [0,1]. As

shown by curve 3 in Fig. 6 b, the results can still convey

useful information for the control property of the system,

although the exact mechanism of enzyme kinetics and/or

regulation might have a statistically significant impact on

the distribution of the control coefficients. The proposed

computational methodology offers the possibility to compu-

tationally investigate the effects of different enzyme kinetics

by assuming alternative kinetics expressions, or by explicit

randomization of the elasticities, and comparing the distribu-

tions of the control coefficients. Such studies will further

provide suggestions on the importance of knowledge of

enzyme kinetics in deriving conclusions about rate-limiting

enzymes and metabolic system responses to changes in

system parameters.

Glycolysis pathway

The glycolysis pathway is the main pathway of glucose

catabolism in most of the organisms, and is one of the most

complex metabolic pathways. In glycolysis, a molecule of

glucose is degraded to two molecules of pyruvate via a series

of enzyme-catalyzed reactions. ATP and NADH are the

carriers of energy and redox, respectively. Under anaerobic

conditions, pyruvate is further transformed into reduced

products such as ethanol and lactic acid. This fermentation

process is one of the major energy sources for living or-

ganisms under respiration-limited conditions. We studied

the anaerobic metabolism in yeast to illustrate the application

of our framework in such a complex metabolic network.

We based our study on a kinetic model of glycolysis of

nongrowing, anaerobic S. cerevisiae developed by Teusink

et al. (2000) through the combination of in vitro and in vivo

experiments. The model, represented by the pathway shown

in Fig. 7, consists of 19 metabolic fluxes (12 reversible and 7

irreversible fluxes) and 17 intermediates including energy

and redox cofactors. In addition to the primary product of

ethanol, byproducts such as glycerol and succinate are also

considered in the system (see Supplementary Material for

model details).

Considering the mass balance equations around each

metabolite node, a 173 31 stoichiometric matrix N was

constructed, and two conserved moieties were identified in

the system: the pyridine nucleotides ([NADH] 1 [NAD] ¼[AN]) and the adenylates ([ATP] 1 [ADP] 1 [AMP] ¼[AP]). The total concentrations of these moieties, [AN] and

[AP], were accordingly included into the analysis as

additional parameters (described in Methods). NAD and

AMP were selected as the dependent metabolites and the

reduced stoichiometry matrix, NR, was constructed. Based

on the pathway stoichiometry and physiological measure-

ments of product formation rates (Teusink et al., 2000), a

steady-state flux distribution was determined by MFA (see

Supplementary Material for details).

FIGURE 7 Anaerobic glycolytic pathway model of nongrowing yeast,

S. cerevisiae, with glucose as the sole carbon source. Chemical species: Gin,

intracellular glucose; G6P, glucose-6-phosphate; F6P, fructose-6-phos-

phate; FdP, fructose 1,6-diphosphate; GAP, glyceraldehydes-3-phosphate;DHAP, dihydroxy acetone phosphate; BPG, bisphosphoglycerate; 3PG,

3-phosphoglycerate; 2PG, 2-phosphoglycerate; PEP, phosphoenolpyruvate;

PYR, pyruvate; AcAld, acetaldehyde; ETOH, ethanol; ATP, adenosine

triphosphate; ADP, adenosine diphosphate; AMP, adenosine monophos-

phate; NAD and NADH, nicotinamide adenine dinucleotide. Pathway steps

and enzymes (in bold): trans, glucose cross-membrane transport; HK,

hexokinase; PGI, phosphoglucose isomerase; PFK, phosphofructokinase;

ALD, fructose 1,6-diphosphate aldolase; TPI, triose phosphate isomerase;

GAPDH, glyceraldehydes-3-phosphate dehydrogenase; PGK, phosphoglyc-

erate kinase; PGM, phosphoglycerate mutase; ENO, enolase; PYK, pyruvate

kinase; PDC, pyruvate decarboxylase; ADH, alcohol dehydrogenase;

ATPase, net ATP consumption; and AK, adenylate kinase.



The elasticities with respect to metabolites are calculated

based on the enzyme kinetics mechanism provided by the

model (Teusink et al., 2000) (see Supplementary Material for

details) and the scaled metabolite concentrations generated

by uniformly sampling enzyme saturation degrees. For the

calculation of weight matrix indices, ATP was chosen as the

independent metabolite of the adenylates moiety and its

fraction in the adenylates pool was sampled based on the

uniformly randomized energy charge within its physiolog-

ically measured range, [0.8–0.9] (Ball and Atkinson, 1975),

and Eq. 33. The catabolic reduction charge is randomized

within experimental range, [0.001–0.0025] (Holzer et al.,

1956). After a Monte Carlo simulation, 5495 random

datasets were sampled before the 5000 sample sets, yielding

stable solutions (91% stability), and the distribution of the

flux control coefficients was determined.

In the yeast anaerobic metabolic pathway, the enzyme

alcohol dehydrogenase (ADH) catalyzes the reaction step

leading to ethanol production and the flux through this

enzyme also reflects the overall flux through the glycolysis

pathway since it is the main output flux of the carbon through

glycolysis. We analyzed the distribution of the control

coefficients of the flux through this enzyme with respect to

three key enzymes: phosphofructokinase (PFK), pyruvate

kinase (PYK), and the enzyme facilitating the transport of

glucose across the plasma membrane (transporter). A large

number of enzymes have been identified as hexose trans-

porters. In this study, as in most of the modeling studies, we

lump the function and kinetics of each enzyme into a single

step. PFK and PYK catalyze phosphoryl group transfer reac-

tions during the glucose breakdown and are major points of

regulation in glycolysis (Fig. 7). Similarly, glucose transport

has been shown to exert considerable control on the glycolytic

flux (Reijenga et al., 2001). Hence, a detailed study of the

control coefficients of ADH with respect to these three key

enzymes would be of interest for a better understanding of the

control of the glycolytic fluxes.

Our result confirmed that glucose uptake exerts the stronger

control on the glycolytic flux, with PFK, and to a lesser extent

PYK, sharing the control (Fig. 8 a). This implies that the

changes in the activity of glucose transporters might lead to

significant changes in anaerobic ethanol production. How-

ever, the mean values of the control coefficients are relatively

low (0.48 for transport, 0.27 for PFK, and 0.09 for PYK),

suggesting that overexpression of any of the enzymes alone

might not lead to significant changes. This is consistent with

experimental studies that have shown that single enzyme

overexpression cannot increase glycolytic flux in yeast

(Schaaff et al., 1989). Furthermore, our results suggest that

simultaneous overexpression of PFK and PYK will also be

ineffective in increasing glycolytic flux, since the probability

of these enzymes, together, to have a flux control coefficient

.0.2, is,0.25. These conclusions are also in agreement with

the previously reported experiments on simultaneous over-

expression of PFK and PYK (Schaaff et al., 1989). Although

these experimental studies have been performed under

conditions different from those used to quantify our Monte

Carlo model, the statistical nature of our framework allows us

to draw some broader conclusions since it could account for

the uncertainties introduced due to the relative differences

between the two experimental systems.

In addition, we studied the control over ethanol production,

and glycolytic flux, exerted by the conserved moieties. As

indicated by Fig. 8 b, increasing the total level of pyridine

nucleotides will likely lead to a decrease of ethanol pro-

duction, whereas increasing adenylates concentration is more

likely to result in an increase of ethanol production. However,

the negative values for some of the control coefficients of the

ethanol productionwith respect to the total level of adenylates

(Fig. 8 b) suggest that there exist some configurations of

elasticities that could lead to a decrease in ethanol production

when the total level of adenylates is increasing. Although

there is no experimental evidence of such coupling between

glycolytic flux and conserved moieties, it illustrates the

potential control of the levels of the conserved moieties on

metabolic fluxes as previously demonstrated for simpler

model systems (Kholodenko et al., 1994).

For many biological organisms under investigation,

systematic and detailed descriptions of enzyme kinetic

mechanisms are absent. This is a typical challenge when

genome-scale metabolic networks are considered (Forster

et al., 2003). To examine the possibility of applying our

framework to such systems, we assumed no information of

enzyme kinetic expressions of the glycolysis model and

randomized all elasticities with respect to metabolites by

uniform sampling between [0,1] for substrates and activator

metabolites, and [�1,0] for products and inhibitor metabo-

lites. In this scenario, we observed similar patterns of control

coefficients distributions (Fig. 8, c and d) as before (Fig. 8,

a and b). However, the distribution of the glycolytic flux

control coefficients with respect to PYK and PFK appear to

be sensitive to the knowledge about the details of enzyme

kinetics, whereas the distribution of the control coefficient

with respect to glucose transport remains almost the same.

These observations suggest that accurate estimation of the

control coefficients of glycolytic flux, with respect to PYK

and PFK, will require knowledge of the kinetic mechanism

of some of the enzymes in the pathway, while glucose

transport is, indeed, one of the most important determin-

ing steps of the glycolytic flux. A more detailed analysis

of the effects of the knowledge of the kinetic mechanisms of

the individual enzymes could help us identify which are the

enzymes for which additional knowledge about their kinetics

is required.

CONCLUDING REMARKS

The statistical and computational MCA framework presented

here combines principles and methods from biochemistry,

mathematical modeling, systems engineering, computational

3760 Wang et al.


biology, and statistics. The framework expands the useful-

ness of MCA in identifying, quantifying, and ranking the

rate-limiting enzymes in complex metabolic networks

through statistical evaluation that accounts for the un-

certainty involved on the parameters that underlie the value

of the flux control coefficients.

In the investigation of biological systems, experimental

approaches are limited to the study of relatively small parts

of a metabolic network. In contrast, a mathematical model

does not adhere to such limitations, and appears to be more

advantageous in studying the behavior of large networks.

Metabolic flux analysis (MFA) allows the study of large

networks but it does not allow the investigation of the net-

work responses to changes in the kinetic parameters of

the network. By combining information from MFA with

the MCA kinetic description, the Monte Carlo MCA frame-

work developed in this article successfully adopts the merits

of both methods, and allows us to infer the global regulation

of cellular metabolism using large-scale computations.

The proposed method allows us to study the network

properties under uncertainty. Such uncertainty may arise

from partial knowledge about the kinetic properties of the

enzymes in the network, as well as from variability in the

environmental conditions under which the in vitro and in

vivo experimental studies are performed. Thus, the analysis

of the simulated results could allow the reconciliation of

experimental information about the same system from

different sources and experimental conditions.

The case studies presented here illustrate that the response

of metabolism to changes in cellular and environmental

parameters depends on the values of the metabolic fluxes and

possibly on the kinetic mechanisms of the enzymes. The

sensitivity of the distribution of the control coefficients on

the kinetic mechanisms used for the estimation of the

elasticities, suggests that uncertainty in the reaction mech-

anism and in the value of the elasticities of some enzymes,

contribute significantly to the uncertainty of control coeffi-

cients. Simulation studies using alternative kinetic mecha-

FIGURE 8 CCDFs of the control coefficients of flux through ADH with respect to glucose transport, PFK, and PYK activity. (a) Control of ethanol

production by three enzymes, transporter (dotted line), PFK (dashed line), and PYK (solid line), based on the kinetic mechanisms provided by Teusink et al.

(2000). (b) Control of ethanol production by the conserved pyridine nucleotides moiety (solid line) and adenylates moiety (dashed line) based on the kinetic

mechanisms provided by Teusink et al. (2000). (c) Control of ethanol production by three enzymes, transporter (dotted line), PFK (dashed line), and PYK

(solid line), for unknown kinetic mechanisms. (d) Control of ethanol production by the conserved pyridine nucleotides moiety (solid line) and adenylates

moiety (dashed line) for unknown kinetic mechanisms.



nisms will help us identify the enzymes whose kinetics

have the most significant effect on the uncertainty of the

predicted system properties. To address this problem

quantitatively we have developed a framework, based on

uncertainty propagation methodologies, that identifies the

relative contribution of the uncertainties in the elasticities of

each enzyme to the uncertainty in the estimation of the

control coefficients (F. Mu and V. Hatzimanikatis, un-

published). The enzymes with the greatest contribution in the

uncertainty of the control coefficients should be the subject

of experimental studies that will determine the details of their

kinetic mechanism for an accurate prediction of the network

properties.

Conserved moieties, enzyme channeling, futile cycles,

cofactor coupling, and regulatory interactions contribute

significantly to the complexity of metabolic networks.

Application of the proposed framework and the related

framework for the quantification of uncertainty propagation

(F. Mu and V. Hatzimanikatis, unpublished) to such complex

pathways will allow us to build a better understanding on

how these elements, e.g., levels of conserved moieties and

enzyme channeling, contribute to the values of the control

coefficients and to the uncertainty in estimating these values

since the framework requires no explicit knowledge of

kinetic parameters.

The MCA formalism allows the estimation of the response

of metabolic networks to relatively small changes in the

metabolic parameters. Estimation of the response of metab-

olism to large changes in the metabolic parameters will

require the use of detailed nonlinear models for the kinetics of

the enzymes in the pathway. However, at the vicinity of small

changes in the metabolic parameters, the results of both

formalisms will be equivalent and the distribution of the

control coefficients will remain unaffected, unless the non-

linear model operates near critical bifurcation points. An

MCA framework, like the one presented here, will provide

guidance for studies of nonlinear models under large changes

in the kinetic parameters. The ability of our framework to

capture the effect of the hypothesized kinetic mechanisms,

and the functional form of the corresponding elasticities, on

the estimated distributions of the control coefficients will

further provide a systematic method and a starting point for

the evaluation of the effects of uncertainty in the kinetic

mechanisms of the enzymes in nonlinear models.

The identifications of drug targets in metabolic diseases,

the metabolic engineering of industrial organisms, and the

identification of the genotype-to-phenotype relationship, are

some of the important applications of MCA (Cascante et al.,

2002). All of these systems involve a great degree of

uncertainty, they are subject to wide variations of their

extracellular environment, and their ultimate function is the

result of a population of individual cells. The ability of the

proposed framework to quantify system properties based on

a range of system parameters is ideal for studying the

properties of multicellular complex systems.

SUPPLEMENTARY MATERIAL

An online supplement to this article can be found by visiting

BJ Online at http://www.biophysj.org.

The authors are grateful for the financial support provided by the

Department of Energy (DE-AC36-99GO103), the National Aeronautics

and Space Administration (NAG 2-1527), and DuPont through a DuPont

Young Professor Award (to V.H.). L.W. received partial support by the

Chinese Government through the State Excellence Scholarship program for

students studying overseas.

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